Interaction-induced doublons and embedded topological subspace in a complete flat-band system

In this work, we investigate effects of weak interactions on a bosonic complete flat-band system. By employing a band projection method, the flat-band Hamiltonian with weak interactions is mapped to an effective Hamiltonian. The effective Hamiltonian indicates that doublons behave as well-defined quasiparticles, which acquire itinerancy through the hopping induced by interactions. When we focus on a two-particle system, from the effective Hamiltonian, an effective subspace spanned only by doublon bases emerges. The effective subspace induces spreading of a single doublon and we find an interesting property: The dynamics of a single doublon keeps short-range density-density correlation in sharp contrast to a conventional two-particle spreading. Furthermore, when introducing a modulated weak interaction, we find an interaction-induced topological subspace embedded in the full Hilbert space. We elucidate the embedded topological subspace by observing the dynamics of a single doublon and show that the embedded topological subspace possesses a bulk topological invariant. We further expect that for the system with open boundary the embedded topological subspace has an interaction induced topological edge mode described by the doublon. The bulk-edge correspondence holds even for the embedded topological subspace.

Figure 1

Creutz ladder model: The yellow circle represents a unit cell, and the blue and magenta shaded objects are compact localized states, $W_{+,j}$ and $W_{-,j}$.

Figure 2

(a) The entire energy spectrum for two-particle system. We see three bands. Here we set $L=20, U=0.1$. (b) The closeup of the lowest band spectrum. Between the 170th and 209th states, a mini dispersive band structure appears, where some extended doublon eigenstates $|{\varphi }_{\ell }^B\rangle$ are lurking.

Figure 3

Flux insertion response: (a) non-flat-band case, $t_0=1.5, U=0.1$, and (b) flat-band case, $t_0=1, U=0.1$. The orange lines represent the eigenstates constituted only by the doublon bases $|B_j\rangle$, that is, $|{\varphi }_{\ell }^B\rangle$. The number of the orange spectrum lines is $L$.

Figure 4

(a) Time evolution of doublon density for flat-band case ($U=0.1, t_0=1$). The initial state is $B_{j_0}|0\rangle$. (b) The dynamical behavior of the eigenvalue of the doublon projector, $\langle \mathrm{\Psi }\left(t\right)\left|P_B\right|\mathrm{\Psi }\left(t\right)\rangle$ for flat-band case. (c) The dynamical behavior of the density-density correlation $C(r,t)$ for flat-band case. (d) Time evolution of doublon density for non-flat-band case ($U=0.1, t_0=1.08$). The initial state is $B_{j_0}|0\rangle$. (e) The dynamical behavior of the eigenvalue of the doublon projector, $\langle \mathrm{\Psi }\left(t\right)\left|P_B\right|\mathrm{\Psi }\left(t\right)\rangle$ for non-flat-band case. (f) The dynamical behavior of the density-density correlation $C(r,t)$ for non-flat-band case.

Figure 5

(a) The entire energy spectrum; we see three bands. Here we set $L=20, U_e=0.1$, and $U_e=0.05$. (b) The closeup of the lowest band spectrum. The band splits to small subbands due to the modulated on-site interaction. The small band gap originates from the doublon subspace.

Figure 6

Time evolution of doublon density for flat-band (a) and non-flat-band ($t_0=1.1$) (b) cases. (c) The behavior of the doublon projector, $\langle \mathrm{\Psi }\left(t\right)\left|P_B\right|\mathrm{\Psi }\left(t\right)\rangle$. (d) Time evolution of the doublon MCD for various parameter cases.

Figure 7

(a) Disorder-averaged expectation value of $P_B$ during time evolution. (b) Disorder-averaged doubon $\mathrm{MCD}$. We set $L=20$, and used 300 quench disorder samples.

Figure 8

(a) The entire energy spectrum; there are five bands. (b) The lowest band spectrum. The band splits to the mini subband due to the modulated on-site interaction. Within the band gap, there appears a single edge mode (the red circle) described by the doublon. The inset shows the doublon density for the edge mode. Here, we set $L=20, U_o=0.1$, and $U_e=0.05$.

Figure 9

Distribution of eigenvalues of the doublon projector $P_B$. (a) Flat-band and uniform interaction, $t_0=1, U=0.1$. (b) Flat-band and modulated interaction, $t_0=1, U_o=0.1, U_{=}0.05$. (c) Dispersive band and uniform interaction, $t_0=1.08, U=0.1$. For all cases, $L=20$ two-particle system, and we display the states in the lowest band.

Figure 10

(a) Two-particle energy spectrum under the change of $|t_0-t_1|$. (b) Behavior of the squared sum of $P_B$. For both cases, we set $L=20, t_0=1, U_o=0.1$, and $U_e=0.05$.

Figure 11

Dynamics of $P_B$ for $U=0.1$ and various disorder strength. We set $L=20$, and averaged over 100 quench disorder samples.